On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation

TL;DR

This paper investigates the existence of bounded positive stationary solutions for a nonlocal Fisher-KPP equation in higher dimensions, using a novel perturbation approach in weighted Sobolev spaces, extending previous results.

Contribution

It introduces a new perturbation method in weighted Sobolev spaces to establish solutions for the nonlocal FKPP equation in , generalizing prior one-dimensional findings.

Findings

Existence of bounded positive stationary solutions in  for small nonlocality.

Development of a perturbation approach applicable to nonlocal evolution equations.

Potential for extending results to hyperbolic invariant sets and other nonlocal models.

Abstract

We study the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et {al.} [Nonlinearity 22 (2009), {pp.}~2813--2844] where the nonlocal FKPP equation has been studied and it was shown for the spatial domain $\mathbb{R}$ andsufficiently small nonlocality that there are only two bounded non-negative stationary solutions. Here we provide a similar result for $\mathbb{R}^d$ using a completely different approach. In particular, an abstract perturbation argument is used in suitable weighted Sobolev spaces. One aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality, {i.e.}, to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not.